on the quasilinear wave equation: utt − Δu + f(u, ut) = 0 associated with a mixed nonhomogeneous...

Nonlrneor Anolysrs, Theory, Merhods & App,pl;mfions, Vol. 19, No. 7. pp. 613-623, 1992. Printed in Great Britam.

0362-546X192 $5.00 + 00 4:~’ 1992 Pergamon Press Ltd

ON THE QUASILINEAR WAVE EQUATION: u,, - Au + f(u, u,) = 0 ASSOCIATED WITH A MIXED NONHOMOGENEOUS CONDITION

NGUYEN THANH LONG? and ALAIN PHAM Ncoc DINH$

t Ecole Polytechnique, Dkpartement de Mathematiques, Ho-Chi-Minh Ville, Vietnam and $ DCpartement de Mathkmatiques, Universite d’OrlCans, BP 6759, 45067 Orleans Cedex, France

(Received 15 October 1990; received in revised form 12 September 1991; received for publication 8 November 1991)

Key words and phrases: Nonlinear Volterra integral equation, local and global existence.

1. INTRODUCTION

WE STUDY the following initial and boundary value (i.b.v.) problem:

utt - Au + f(u, u,) = 0; O<x<l,O<t<T (1.1)

u,(O, t) - hu(O, t) = g(t); h constant > 0

u(1, t) = 0 (1.2)

4x, 0) = u,(x); 4(x, 0) = u,(x). (1.3)

In [3], Ficken and Fleishman established unique global existence and stability for the equation

ll XX - u,, - 2cyiu, - (YOU = eu3 + b, E > 0 small. (1.4)

Rabinowitz [5] has proved the existence of periodic solutions for

u,, - UXX + 2cYiu, = &f, (1.5)

where E is a small parameter and f is periodic in time. Also relevant to our considerations is the paper by Caughey and Ellison [ 11, in which a unified approach to these previous cases is presented to discuss the existence, uniqueness and asymptotic stability of classical solutions for a class of nonlinear continuous dynamical systems. In [2], Ang and Pham established unique global existence for the i.b.v. problem (l.l)-(1.3) with f(u, u,) = ju,la sgn(u,), (0 < CY < 1) and h = 0. In this latter case this problem governs the motion of a linear visco- elastic bar. The aim of this paper is to give a generalization of [2] and to prove for convenience assumptions on f a local existence theorem for h > 0. If we call u,,(x, t) such a solution, this solution depends continuously on the parameter h and we give sufficient conditions for global existence.

2. EXISTENCE THEOREM

We first set some notations:

Lq = L4(sz), N’ = ffyn), fi = (0, I), Q = C2 x (0, T)

where H’ is the usual Sobolev space on 0. Let ( *, * > denote either the L* inner product or the pairing of a continuous linear functional with an element of a function space. The L2 norm will

613

614 NGUYEN THANH LONG and A. PHAM NGOC DINH

be denoted by I/ * 11. Let II - Jlx b e a norm on a Banach space X and let X* be its dual space. We denote by Lp(O, T; X), 1 i p _ < co, the space of real functions f: (0, T) + X with

ilf IILpcO,T;Xj = (~oTiifWkGdr>‘lp < 03 for 1 5 P < ~0

Ilf II L-(O,T$) = e;; ;.tp llf(~)llx forp = co.

Let V = {V-E H’ / v(l) = 0)

‘I a(u, u) = <au/ax, au/ax> =

I au/ax. au/ax do. (2.1)

II 0

V is a subvector space of H’ and on I/ (a(v, v))“‘~ is a norm equivalent to llvllH~. We will set

II4H1 = (4v, 4Y2. We shall make the following assumptions:

O<h<h, (2.2)

uo E H’, u1 EL2 (2.3)

g E ff’(O, 7-1 vT>O; g(0) exists (2.4)

f: R2 + R (2.5) satisfies the following conditions:

(i) f(0, 0) = 0

(ii) (f(u, ~7) - f(4 3)) * (u? - @I 2 0 v 4 v, U, E R

there exists 01, /3 constants 0 < 01 < 1, 0</35 1 (2.6)

and two functions B, , B,: R, + R, continuous, nondecreasing and such that: (i) B,(luj) E L2’(‘-“) (Q), vu E L”(0, T; V), v T > 0

(ii) B2(ld) E L'(Q) v v? E L2(Q) (iii) If@, v) - f(u, @)I 5 B,(ld)b - @I”, vu, P, ul E R (iv) If@, u?) - f(& @I 5 B2(ld)lu - tile, vu, 6, v, E R.

We write u’(t) := adat = u,; u”(t) := a2u/at2 = utf. We then have the following theorem.

THEOREM 1. Let (2.3)-(2.6) hold. Then there exists T > 0 such that the i.b.v. problem (1 .l)-( 1.3) has at least a weak solution u on (0, T) such that

u E L”(0, T; I’), u, E L”(0, T; L’).

If p = 1, then the problem (1 .I)-(1.3) has exactly one solution.

Proof. The proof consists of several steps.

Step 1. The Galerkin approximation. Consider a special orthonormal basis on V’.

Wj(X) = J2/(1 + /1/‘) . COS (JLJX)) Aj = (2j - 1)X/2, jEN*

formed by the eigenfunctions of the Laplacian A.

Quasilinear wave equation 615

Let the subspace (wt , w2, . . . , w,) be generated by the distinct basis elements w, , w2, . . . , w, of I/. Put

u,(t) = i 5j,(flwj> (2.7) j=l

where tj,, satisfy the following system:

<ui(t), wj> + a(un(r), wj) + thu,tO, t, + gCt)) . wj(“) + (f(“n(t)3 uA(t))f wj> = O3

j = 1,2, . . ..n

u,(O) = %,1 = ~ ~j~ Wj j UO in H’ (2.8) j=l

II

u;(o) = Uln = C PjnWj + u1 in L2. j=l

It follows from the hypotheses of the theorem that system (2.8) has a solution on an interval [0, T,]. The following estimates allow one to make T, = T for all n.

Step 2. A priori estimates. Multiply each equation in (2.8) by <jl,(r), sum up with respect to j. We then have

+ d/dt S,(t) + (f(u,,(t), u;(t)) - f(u,(t), 0), u;(t)>

where = -g(t)u;(O, 0 - (f(u,(t), O), u;(t)> (2.9)

S,(f) = IlUt)l12 + 11%1(~)11: + ~l~,(O9 o12. (2.10)

By assumptions (2.4) and (2.5) and after integrating with respect to the time variable from 0 to t we have:

‘I

S, 5 S,(O) + 2g(ON,,(O) - %(t)u,(O, 0 + 2 I

g’(W,(O, 4 dt I_ 0

- 2 I ” <f(u,(r), 01, u,‘Js)) ds. (2.11)

,O

Since -u(x, t) = ji u&s, t) dt for every u E I/, then the embedding constant Co for V G e’(Q) can be taken as 1. Thus, from (2.11) we get

where

-1

s,(t) 5 g(t) + I

%,(@ dt + 4 -0 !

“I 11.%(7), O)ll . iI&,,d dt (2.12) 0

g(t) = 2C, + 4g2(t) + 4 ! “’ ]g’(r)l’ds. ,O

C, being the constant defined by

S,(O) + 21g(0) * ~OAW 5 c, . In deriving (2.1 l), we have made use of the inequality

2ab 5 a2/cr + cxb2 for a, b 2 0 V n! > 0. (2.13)

616 NGUYEN THANH LONG and A. PHAM Ncor DINH

Hypothesis (2.6iv) implies that

l.W#), 0)l 5 &(O)l~,~(W 5 &(0)IWW2. (2.14)

Since H’(0, T) G C’[O, T] the function g(t) is bounded a.e. on [0, T] by a constant M, depending on T. Therefore it follows from (2.12) and (2.14) that

where

S,,(r) I M, + I “&S,(r)) dr, Ost<7;,sT

,, 0

k”(s) = s + 4B2(0)sl+3’2.

(2.15)

The function &s) is nondecreasing for s 2 0, hence

s,(t) 5 s(t) v t E [0, T] for each T > 0. (2.16)

S(t) being the maximal solution of the nonlinear Volterra integral equation with nondecreasing kernel [4] on an interval [0, T], equation given by

S(r) = M, + I“ &S(r)) dr. (2.17)

Now we need an estimate on

I ” lu;(O, s12 ds. .O

The coefficient lj,(t) of u,(t) satisfies the system of ordinary differential equations

<J?I(t) + nj”<j,(r) = -141wjI12[<f(un(t)~ uii(t))t wj> + thunto, f) + g(t)) . wj(“)l

ljn(O) = ajn

rJx3 = p/n . System (2.18) is equivalent to the following:

rj,(t) = ~j~ COS(A,t) + @jn(Sin(Itjt)/Jj) - 1//l~jl(2

’ I ” (sin(Aj(t - $/3Lj)[(f(Un(f), U;(T)), wj)

.O

(2.18)

+ (hu,(O, T) + g(r))Wj(O)] ds. (2.19)

Put

K,,(f) = c sin JLJt/jLJ (2.20) j=l

Y,(t) = i wjm CYJn COS(~jt) + /Ij,(Sill(2jt)/Itj)] j=l

- J2 i i’ (sin Aj)(f - r)/Ajzj(f(u,(r), u;(r)), W,/I(WjII> ds (2.21) j=l .O

d,(l) = 2h K,(t - r)u,(O, t) dr. (2.22)


Then ~~(0, f) can be rewritten as

~~(0, t) = y,(t) + s,(t) - 2 K,,(t - r)g(r) dr. (2.23)

We shall require the following lemmas.

LEMMA 1. There exists a positive continuous function D(t) independent of n and a constant Cz such that

~ 1 tvMi2 dr 5 G + o(t) 1: lIf(u,(7),~~(5))ll~d~ i

v t E [0, T] for any T > 0. (2.24)

The proof of lemma 1 can be found in [2].

LEMMA 2. ,af ns

1 II

2

K;(s - T)g(T) dr ds 5 A4; v t E [0, T] for any T > 0 (2.25) .0 ..o

M: indicating a constant depending on T.

Proof. Integration by parts gives

‘> t

I

I’ f K,‘,(t - r)g(r) dr = g(O)K,(t) +

1 K,(t - r)g’(t) dr. (2.26)

I, 0 Next, by (2.26)

‘1 ‘s

! I! K;(s - r)g(r)dr,‘ds 5 21;K&9)d0Y ,g2(0) + Tj:g”(i)dr]. (2.27)

.0 .0

Lemma 2 is proved since K,(t) converges strongly to a function K(t) in L2(0, T) for any T > 0.

LEMMA 3. There exists two constants M$ and M+ depending on T such that

!

sf ” t IdA( dr 5 M; + h2TM;

I (u;(O, r)12 dr V t E [0, T], for any T > 0. (2.28)

.0 .O

Proof. By (2.22) we get

6;(t) = 2hu,,(O)K,,(t) + 2h K,(t - r)u;(O, t) dr. (2.29)

From (2.29) we easily derive, after some rearrangements,

! ” (S;(s)(2 ds I 8h2 ,~K:(B)d& [u&(O) + T,[; (u;(O, r)/‘drj.

I (2.30)

.0

Finally we obtain

i ’ lS;(s)/2 ds 5 8C, hM; + 8h2TM;

i ’ ju;(O, r)12 dr (2.31)

,0 .0

since h&(O) I S,(O) I C, V n and ScKi(@ d0 5 M: constant depending on T. Now we are ready to state an a priori estimate for S,’ luA(O, 7)12 ds.

618 NGCJYEN THANH LONG and A. PHAM NGOC DINH

LEMMA 4. There exists T > 0 such that ji’ lu;(O, s)12 dr I MS.

Proof. By (2.23) we have

uA(O, I) = y:,(t) + s;(t) - 2 \“K&@ - s)g(r) ds.

From (2.32) we get .lo

I

aI “f lu:,(0,s)/2ds I 3

,0 I ” /Y:,(s)12ds + 3 )

I 0 I 0 I&(s)/2ds + 12 1’

I 0

It follows from lemmas l-3 and (2.33) that ,I I ( I’f

II as

K;(s - s)g(r) dr 2 ds. (2.33) .0

(2.32)

I l&(0, s)/‘ds 5 3C2 + 3W) llf(U4, &(~)ll~ ds .o I .O

\ ” f 3M; + 12M; + 3h’TM; l&(0, r)12 dt. (2.34)

,0

If we choose T such that 3h’TM: 5 4, then (2.34) yields

I ”

,O I&,(0, s)12 ds S M; + 6D(f) [’ 11 f(u,,(T), uL(t))l12 ds.

I 0 (2.35)

On the other hand, by assumption (2.6) we have

IlfGM)~ UO)l12 5 W(Il~,(f)llv) * ll~:,(m~~“(*, + 2~2mwl/$~.

Note that I/. IIL~crcRj 5 1) * Ilt(i(n,. Hence

IlfoM), 4W)l12 5 mJwww + wm9?a

Finally from (2.35) and (2.37) it follows that there exists T > 0 such that

\ “l~;~(0, t)12 dr 5 M;.

,, 0

(2.36)

(2.37)

(2.38)

Step 3. The limiting process. By (2.16), (2.37) and (2.38) (u,) has a subsequence still denoted (u,) such that

u, --* u in L”(0, T; V) weak* (2.39)

U:, + U’ in L”(0, T; L2) weak* (2.40)

&(O, 0 + u(O, f) in L”(0, T) weak* (2.41)

Uk(O, t) + U’(0, t) in L2(0, T) weak (2.42)

f@,> UA) + x in L”(0, T; L2) weak*. (2.43)

By (2.16) and (2.38) on the one hand, and by (2.39) and (2.40) on the other, we can extract from {u,) a subsequence still denoted by (u,) such that

u,(O, r) + u(O, 0 uniformly in CO([O, T]) (2.44)

u, + U strongly in L2(Q). (2.45)


If we pass to the limit in the equation (2.8) we find without difficulty from (2.39), (2.40), (2.43) and (2.44) that u(t) satisfies the equation

d/dt(u’(t), u> + a(u(t), u) + (hu(0, t) + g(0)@) + Q(t), U> = 0, for any u E V. (2.46)

Since U, U, E C’(O, T; L’), we have u,(O) + u(0) strongly in L’. Thus,

U(0) = I_&). (2.47)

On the other hand, the functions (u;(t), wj) and (u’(r), wj) belong to C’(O, T). Therefore, (u;(O) - u’(O), wj> + 0 for n + co. Hence,

U’(0) = Ur . (2.48)

We shall now require the following lemma.

LEMMA 5. Let u be the solution of the following problem

UU -Au+X=O

u,(O, 0 - WO, f) = g(t); U(1, I) = 0

U(0) = U(J ; 4(O) = UI

with u E L”(0, T; V) and U, E L”(0, T; L’), then we have

$l]tlf(t)~12 + +IIu(t)II$ + i’ (hu(O, T) + g(s))Ut(O, 5) dr + [’ (X(T), U,(T)) dr .I 0 I, 0

1 +A2 + tll&. The proof of lemma 5 can be found in [2].

(2.49)

(2.50)

Remark. If u. = U, = 0 there is equality in (2.50). Next, we claim that

x = f(u, u,). It follows from (2.8) that

(2.5 1)

” f

I <f(~,(s)> u;(r)), U;(T)) dT = ill~~n~~2

,O + tll%,ll; - +lb;(f)il’ - [kbJ:@. T) dT

.O

+ th&(O) - +llMllF - w47(0, f)12. (2.52) By lemma 5 we have

]im_szp ,‘I (.IGM), u;(r)), G(r)) dr 5 HullI !

+ +llu& + +hu,2(0) - *hz2(0, t) ‘f

_ ! g(@u’@, r> dT - +llu(t>ll$ - +llu’(t)l12 .O

(X(T), u’(T)) dT a.e. t E (0, T). (2.53)


By (2.45)

By (2.6) and (2.54) we get u,, + u a.e. in Q.

f(u, 16) + AU> 4) a.e. in Q, V C$ E L’(Q).

Hence, by the dominated convergence theorem, we obtain

(2.54)

(2.55)

(2.56)

From (2.56) we derive that

lim ” I

” r

(f(u,(t), ad), U;(T) - 6(r)) ch = n-m .,o I

<f(u(s)> 4441, U’(T) - +(d) dr v 4 E L2(Q). .O

(2.57)

Next, consider

It follows from (2.43), (2.53), (2.57) and (2.58) that

Thus

0 5 limsupX,(r) ZS /

(X(r) - f(u(r), #(r)), u’(r) - cb(r)> dr. n-m .O

<,: (X(r) - f(u(r), 4(r)), u’(r) - 4(r)) dr 2 0 i

V ~5 E L’(Q).

Let d(t) = u’(t) - lw(t), /1 > 0, w E L’(Q). Then we obtain

We have

since

‘I

I (X(T) - f(U(T), U’(T) - AW(T)), W(T))) dt 2 0 v w E L'(Q). .O

‘f lim

1 (f(u(r), U’(r) - A”‘(r)) - f(u(r), u’(r)), W(r)) dr = 0

x-0, <O

(2.61)

(2.62)

Il&, u, - Aw) - f(‘, u,)lI&(2) S ‘allB,(1UI)IIL2’(1-0)tQ,Ilwll~Z~Q,. (2.63)

Finally, it follows from (2.61) and (2.62) that

Therefore i

f (X(T) - f(U(T), U’(d), W(T)> dT 2 0 v w E L2(Q).

.O

x = f(u, u,) a.e. in Q, as claimed.

(2.59)

(2.60)

(2.64)


Uniqueness

Assume now that p = 1 in (2.6). Let ur, u2 be two solutions of the i.b.v. problem (1.1)~(1.3) and let u = ur - u2. Then u is the solution of the following i.b.v. problem

Utt -Au+x=O, o<x< 1, O<tsT

z&(0, t) - hu(0, t) = 0 (2.65)

u(1, t) = u(0) = u,(O) = 0

u E L”(0, T; I’); u, E L”(0, T; L’); 2 = _I-(4 9 u;) - f(uz 9 W.

By using lemma 5 with u. = u1 = g(t) = 0, we have

311~‘(t)l12 + tllw>ll2,

By (2.66)

‘f

I

t +h ~(0, s)u'(O, 7) dr + (J?(r), u'(7)) dr = 0 a.e. t E (0, T).

JO Jo

lIu’(t)l12 + Mm + hu2(0, T) 5 2 I

li.f(~1(7>, U;(T)) - _f(U7), ui(Q1ll llu’(7)li .O

since the function f(ur , -) is monotone. From the hypothesis (2.6) we deduce that

lI.m, 9 4) - .mz 3 G)ll 5 Il~z(l& I)11 * Ilull v.

Let

a(t) = Ilu’(t)l12 + Ilu(t)112, + hu?O, 0.

By (2.67)-(2.69) it follows that

o(t) 5 2 II&(Iu;(d)ll~(7) d7

(2.66)

dr

(2.67)

(2.68)

(2.69)

(2.70)

i.e. o = 0 by Gronwall’s lemma.

3. CONTINUOUS DEPENDENCE OF THE SOLUTION

In this part we assume that j3 = 1. The problem (1 . l)-( 1.3) according to theorem 1 admits a unique solution u(x, t) = u,(x, t), h > 0. We make the following supplementary assumption on the function B,(e):

B,: L2 + L2, takes bounded sets into bounded sets.

Then we have the following theorem.

(3.1)

THEOREM 2. Let /3 = 1. Let (2.3)-(2.6) and (3.1) hold. Then there exists T > 0 such that the i.b.v. problem (l.l)-(1.3) with h = 0 has a unique solution C E L”(0, T; V) and such that Qt E L”(0, T; L2). Furthermore

lim(Ilh - &~O,~o + II4 - ~‘lIL-cO,T;L~j) = 0. h-0


Proof. Let u,, (resp. u,,) be the solution of the i.b.v. problem (1 .l)-(1.3) with the parameter h(resp. h’). Let w = u,, - uh,. Then w satisfies

Wrt -Aw+%,=O O<x<l,O<t<T

where

w,(O, I) = h . w(0, t) + hz+(O, t)

w( 1) f) = w(x, 0) = w,(x, 0) = 0

w E L”(0, T; V); w, E L”(0, T: L”)

x, = f(u, 4) - f(u, 4)

t’i = h - h'.

(3.2)

(3.3)

Proceeding as in the proof of theorem 1, we deduce that

u (resp. u,) is bounded independently of h in L”(0, T; V) [resp. in L”(0, T; L’)]. (3.4)

Let

o(t) = IIw’(t)l12 + Ilw(dF.

As before we can derive the following inequality:

” a(t) I c,lhl + I /1~2(lG,4~)l)lldd dr. (3.5)

Assumptions (3.1) and (3.4) yield that

lI~,tl~~4~)I)II 5 CA constant depending only on T.

Next, by (3.5) and (3.6) and Gronwall’s lemma,

)I a(t) 5 C+li + C:

I a(s) dr 5 C’$/$ for any t E [0, T].

.0 Thus

(3.6)

(3.7)

lluh - ~,&o,~;V)+l14 - G/~&,,T;Lz) 5 C;lh - h’l. (3.8)

Denote by W the Banach space

W = (u E L”(0, T; V)lu, E L-(0, T; L’)]

with the norm (3.9)

II4Il.v = (Il4Zm(0,7;Y) + II&(O,r;&.

Let (h,] be a sequence such that h, > 0, h, -+ 0 as n + 00. It follows from (3.8) that (u,,?) is a

Cauchy sequence in W. Thus there exists u” E W such that

uh, + u in W strongly. (3.10)

By passing to the limits such as in the proof of theorem 1, we conclude that ~2 satisfies

the equation

d/dt(W,, u) + a(fi(t), u) + g(t)u(O) + ( f(u”,tit), u> = 0 for any u E V’, a.e. t E (0, T)

(3.11)


and the initial conditions G(O) = 240

u”,(O) = 241.

(3.12)

Uniqueness is proved in a standard manner, such as in the proof of theorem 1. Hence, letting

h’ -+ 0 in (3.8) gives

11% - G=,(O,T;V) + II& - Q’llsyO,T;L2) 22 a. m (3.13)

THEOREM 3. Let /3 = 1. Let (2.3)-(2.6) and (3.1) hold. (i) Let h = 0. Then for each T > 0, the i.b.v. problem (l.l)-(1.3) has a unique solution

U* E L-(0, T; V) such that u*’ E L”(0, T; L2). (ii) v E > 0, v T > 0, there exists h = h(&, T) > 0 such that if 0 < h < K then the i.b.v.

problem (1 . l)-( 1.3) has a unique solution uh such that

uh E L”(0, T; V), u;, E L”(0, T; L2)

and satisfying

Proof (if In the case h = 0 we proceed as in the proof of theorem

following a priori bounds:

11~~(~)112 + ~l~~(~)ll~~ 5 MT vr~[O,T],vT>0

i‘T

(3.14)

1 and we derive the

(3.15)

\ lu,(O, t)12 dt i Mr v f E [0, T], v T > 0. (3.16)

,0

Then we can prove in a similar manner, such as in the proof of theorem 1, that the limit ZP of the sequence lu,] defined by (2.8) satisfies equation (1.1) associated with the boundary

conditions ~~(0, t) = g(t), ~(1, t) = 0. (ii) In the proof of theorem 1, if we choose

k = Min(e/C:, lld6T. M:) VE>O,VT>O

(M; defined in (2.28) and C+ in (3.13)), then for 0 < h < k we get the following:

I/% - ~*GmfO,T;V) + llut, - ~*‘l&o,T;L2> < 8.

The proof is complete.

(3.17)

(3.18)

Acknowledgemenl-The authors would like to thank the referee for his corrections and suggestions.

I.

2.

3.

4. 5.

REFERENCES

CAUCHEY T. & ELLISON J., Existence, uniqueness and stability of solutions of a class of nonlinear differential equations, J. m&h. Analysis Appiic. 51, 1-32 (1975). DANG D. ANG & PHAM N. DINH, Mixed problem for semi-linear wave equation with a nonhomogeneous condition, Nonlinear Analysis 12, 581-592 (1988). FICKEN F. & FLEISHMAN B., Initial value problems and time periodic solutions for a nonlinear wave equation, Communspure appl. Math. 10, 331-356 (i957). LAKSHMIKANTHAM V. 81 LEELA S.. Differential and Intenrul heauaiities. Vol. I. Academic Press, New York (1969). RABINOWITZ P. H., Periodic solutions of nonlinear hyperbolichifferential equations, Communs pure app/. Mark 20, 145-205 (1967).

on the quasilinear wave equation: utt − Δu + f(u, ut) = 0 associated with a mixed nonhomogeneous...

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